Degree-inverting involutions on matrix algebras

Let \(F\) be an algebraically closed field of characteristic zero, and \(G\) be a finite abelian group. If \(A=\oplus_{g\in G} A_g\) is a \(G\)-graded algebra, we study degree-inverting involutions on \(A\), i.e., involutions \(*\) on \(A\) satisfying \((A_g)^*\subseteq A_{g^{-1}}\), for all \(g\in G\). We describe such involutions for the full \(n\times n\) matrix algebra over \(F\) and for the algebra of \(n\times n\) upper triangular matrices.